The Prime Field Architecture of the Arithmetic Modulation Field: Yang-Mills Existence and Mass Gap on R⁴ — Volume 6: The Osterwalder-Schrader Axioms

Volume 6 of 14. Establishes the Euclidean axiomatic foundation of the Prime Field Architecture: the Schwinger functions satisfy all five Osterwalder-Schrader axioms OS1-OS5. Defines the Schwinger functions S_n(x₁,...,xₙ) = ⟨Ω₀|φ(x₁)...φ(xₙ)|Ω₀⟩ within the Prime Field Architecture. Proves Theorem 23 (OS1 Temperedness): S_n bounded by exp(−m_gap·r)·r−3/2. Proves Theorem 24 (OS2 Euclidean Covariance): translation invariance, bilateral reflection invariance, palindromic rotational symmetry; full SO(4) covariance in the continuum limit. Proves Theorem 25 (OS3 Reflection Positivity): the transfer matrix T_PFA = exp(−a·H_a) is strictly positive; H_a ≥ 0 with spectral gap Δ_E = 1 > 0. Proves Theorem 26 (OS4 Symmetry): field quanta are bosons. Proves Theorem 27 (OS5 Cluster Decomposition): exponential factorization rate exp(−m_gap·|a|). Proves Theorem 28 (Vanishing One-Point Function): S₁ = 0, confirming vacuum purity. Proves Theorem 29 (OS Reconstruction): the Wightman quantum field theory exists, is unique, and has mass gap spec(H_W) ∩ (0,m_gap) = ∅.